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A373848 Numbers k such that k is not divisible by p^p for any prime p, and for which 1 < A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

%I A373848 #21 Jun 22 2024 18:02:38
%S A373848 5,9,15,25,30,42,45,63,75,105,110,125,126,147,150,165,175,198,210,225,
%T A373848 231,245,275,294,315,330,343,363,375,385,441,462,495,525,539,605,625,
%U A373848 650,686,693,726,735,750,770,825,847,875,882,990,1029,1050,1089,1125,1155,1170,1190,1210,1225,1250,1331,1375,1386,1430
%N A373848 Numbers k such that k is not divisible by p^p for any prime p, and for which 1 < A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.
%C A373848 The initial 5 is the only prime in this sequence (for a proof, consider _Henry Bottomley_'s Sep 27 2006 formula for A024451), the next three terms 9, 15, 25 are only semiprimes (see A087112 and A370129), and there are 21 terms with three prime factors in total: 30, 42, 45, 63, 75, 105, 110, 125, 147, 165, 175, 231, 245, 275, 343, 363, 385, 539, 605, 847, 1331 (see A369979, A370138 and A373844). In general, there should be only a finite amount of terms x such that A001222(x) = k, for any k >= 1.
%C A373848 It is conjectured that 5 is the only fixed point of A373842, which would imply that x=6 is the only number for which A003415(x) = A276086(x). See A351228.
%H A373848 Antti Karttunen, <a href="/A373848/b373848.txt">Table of n, a(n) for n = 1..1565</a>
%o A373848 (PARI)
%o A373848 \\ Uses the code from A373842, or its precomputed data:
%o A373848 A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };
%o A373848 isA373848(n) = if(!A359550(n), 0, my(u=A373842(n)); ((1<u) && (u<=n)));
%o A373848 (PARI)
%o A373848 A002620(n) = ((n^2)>>2);
%o A373848 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A373848 A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
%o A373848 \\ The following routine checks that n is not a prime larger than five, is in A048103, and in case n is odd, rules out cases that certainly cannot give A373842(n) <= n:
%o A373848 prefilter_for_A373848(n) = if(n < 3 || (isprime(n) && n > 5), 0, my(f=factor(n), k=#f~, lpf=f[1,1], p=f[k,1], m=f[k,2]); for(i=1, k, if(f[i, 2]>=f[i, 1], return(0))); if(2==lpf, return(1)); while(p>lpf, p = precprime(p-1); m *= p; if(m>n, return(0))); (1));
%o A373848 isA373848(n) = if(!prefilter_for_A373848(n), 0, my(x=A276085(n)); if(x>A002620(n), 0, (!isprime(x) && A003415(x)<=n)));
%Y A373848 Intersection of A048103 with the setwise difference A373847\(A373846 U {1, 2}).
%Y A373848 Subsequence of A373847.
%Y A373848 Cf. A001222, A003415, A024451, A087112, A276085, A276086, A359550, A369979, A370129, A370138, A373842, A373844, A373845.
%Y A373848 Cf. also A351228, A373603.
%K A373848 nonn
%O A373848 1,1
%A A373848 _Antti Karttunen_, Jun 20 2024